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Generalized entanglement temperature and entanglement thermodynamics

Ashis Saha, Sunandan Gangopadhyay, Jyoti Prasad Saha

Published Apr 2, 2020• 0 citations
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Abstract

In this paper, we observe that a Smarr like thermodynamic relation is satisfied by the HEE of a gravity background dual to a nonconformal field theory. Such a relation emerges naturally by demanding that the generalized entanglement temperature ($T_g$) produces the exact Hawking temperature as the leading term in the IR limit ($l\rightarrow \infty$). In the UV limit ($l\rightarrow 0$) the leading term of the generalized entanglement temperature ($T_g$) gives the entanglement temperature which is inversely proportional to the subsystem size $l$. The results reduces to the existing results in the limiting case of the nonconformal parameter going to zero.

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In this paper, we observe that a Smarr like thermodynamic relation is satisfied by the HEE of a gravity background dual to a nonconformal field theory. Such a relation emerges naturally by demanding that the generalized entanglement temperature (TgT_gTg​) produces the exact Hawking temperature as the leading term in the IR limit (l→∞l\rightarrow \inftyl→∞). In the UV limit (l→0l\rightarrow 0l→0) the leading term of the generalized entanglement temperature (TgT_gTg​) gives the entanglement temperature which is inversely proportional to the subsystem size lll. The results reduces to the existing results in the limiting case of the nonconformal parameter going to zero.
We study thermodynamics of entanglement entropy for weakly excited states in certain nonconformal field theories, whose gravity duals are given by nonconformal Dp-branes. We observe that the entanglement entropy of a sufficiently small system in nonconformal backgrounds still obeys a first-law-like relation, just as the anti--de Sitter counterparts investigated in [J. Bhattacharya, M. Nozaki, T. Takayanagi, and T. Ugajin, Phys. Rev. Lett. 110, 091602 (2013)]. The effective temperature is proportional to the inverse of the size of the subsystem. The proportionality is a dimensionless constant which is only determined by the shape of the entangling region and independent of any coupling. This universality is confirmed by working with the 10-dimensional string frame metric as well as the lower-dimensional effective metric. When the entangling region is a strip and translational invariance is broken by metric fluctuations, we derive a first-law-like relation where additional components of the stress energy tensor are involved.
We study the holographic entanglement entropy of nonlocal field theories at both extremality and finite temperature. The gravity duals, constructed by Nozaki et al. [J. High Energy Phys. 10 (2012) 193], are characterized by a parameter www. Both the zero-temperature backgrounds and the finite-temperature counterparts are exact solutions of Einstein-Maxwell-dilaton theory. For the extremal case we consider the examples where the entangling regions are a strip and a sphere. We find that the leading-order behavior of the entanglement entropy always exhibits a volume law when the size of the entangling region is sufficiently small. We also clarify the condition under which the next-to-leading-order result is universal. For the finite-temperature case we obtain analytic expressions in both the high-temperature and low-temperature limits. In the former case the leading-order result approaches the thermal entropy, while the finite contribution to the entanglement entropy at extremality can be extracted by taking the zero-temperature limit in the latter case. Moreover, we observe some peculiar properties of the holographic entanglement entropy when w=1w=1w=1.
Utilizing the holographic technique, we investigate how the entanglement entropy evolves along the renormalization group flow. After introducing a new generalized temperature which satisfies the thermodynamicslike law even in the IR regime, we find that the renormalized entropy and the generalized temperature in the IR limit approach the thermal entropy and thermodynamic temperature of a real thermal system. This result implies that the microscopic quantum entanglement entropy in the IR region leads to the thermodynamic relation up to small quantum corrections caused by the quantum entanglement near the entangling surface. Intriguingly, this IR feature of the entanglement entropy universally happens regardless of the detail of the dual field theory and the shape of the entangling surface. We check this IR universality with a most general geometry called the hyperscaling violation geometry which is dual to a relativistic nonconformal field theory.
Entanglement temperature is an interesting quantity which relates the increased amount of entanglement entropy to that of energy for a weakly excited state in the first-law of entanglement entropy, it is proportional to the inverse of the size of entanglement subsystem and only depends on the shape of the entanglement region. We find the explicit formula of entanglement temperature for the general hyperscaling violation backgrounds with a strip-subsystem. We then investigate the entanglement temperature for a round ball-subsystem, we check that the entanglement temperature has a universal form when the hyperscaling violation exponent is near zero.
In this paper we have studied the effect of deformation and temperature on holographic entanglement entropy and mutual information between two subsystems in a deformed field theory at finite temperature. The TT‾T\overline{T}TT deformation operator in field theory introduces a cutoff in the dual gravity background. We define different regimes of parameters and calculate the entanglement entropy and mutual information analytically. We observe that temperature and deformation parameter have similar effects on entanglement entropy and mutual information. They both lead to decrease in mutual information or somehow the correlation between the two subsystems. Interestingly we observe the emergence of correction terms in entanglement entropy and mutual information that are universal and do not depend on the scale of the nonlocal operator.
Entanglement entropy in local quantum field theories is typically ultraviolet divergent due to short distance effects in the neighborhood of the entangling region. In the context of gauge/gravity duality, we show that surface terms in general relativity are able to capture this entanglement entropy. In particular, we demonstrate that for 1 + 1-dimensional (1 + 1d) conformal field theories (CFTs) at finite temperature whose gravity dual is Banados–Teitelboim–Zanelli (BTZ) black hole, the Gibbons–Hawking–York term precisely reproduces the entanglement entropy which can be computed independently in the field theory.